

A086792


Orders of finite groups G with the property that the sum of the orders of all the proper normal subgroups of G equals the order of G.


1



6, 12, 28, 30, 56, 360, 364, 380, 496, 760, 792, 900, 992, 1224, 1656, 1680, 1980
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

The only Abelian groups with this property are the cyclic groups C_n where n is a perfect number, so this sequence can be seen as a groups analogy of perfect numbers.
Derek Holt (mareg(AT)mimosa.csv.warwick.ac.uk) computed the orders of the nonAbelian groups in the sequence up to n=500 and commented "In general, if 2^n  1 is a Mersenne prime, then 2^(n1)*(2^n  1) is a perfect number and the group with presentation < x,y  x^(2^n1) = 1, y^(2^n) = 1, y^1 x y = x^1 > has order equal to the sum of the orders of its proper normal subgroups." So if n is an even perfect number, 2n also belongs to this sequence (the numbers 12 and 56 above).


LINKS

Table of n, a(n) for n=1..17.
Tom Leinster, Perfect numbers and groups
Tom De Medts, Attila MarĂ³ti, Perfect numbers and finite groups
Tom De Medts, Leinster groups
Sci.math, Groups analogy of perfect numbers


CROSSREFS

Cf. A000396.
Sequence in context: A009242 A032647 A327165 * A064987 A057341 A068412
Adjacent sequences: A086789 A086790 A086791 * A086793 A086794 A086795


KEYWORD

nonn,more


AUTHOR

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 04 2003


EXTENSIONS

a(10)a(17) added using "Leinster groups" link by Eric M. Schmidt, May 02 2014


STATUS

approved



